Question: Zach scores a ringer $40\%$ of the time that he throws a horseshoe. Let $R$ be the number of throws it takes Zach to score his first ringer in a game. Assume the results of each throw are independent. Find the probability that it takes Zach $5$ or more throws to score his first ringer. You may round your answer to the nearest hundredth. $P(R\geq 5)=$
Explanation: Without a fancy calculator On each throw: $P({\text{ringer}})=0.4$ $P(\text{no ringer}})=0.6$ If it takes Zach $5$ or more throws to score his first ringer, then does not make a ringer on his first $4$ throws. $\begin{aligned} P(R\geq5)&=P(\text{no ringer first 4}) \\\\ &=(0.6})^4 \\\\ &= 0.1296 \end{aligned}$ $P(R\geq 5) = 0.1296 \approx 0.13$